where \({R}(s,s^{\prime }) = {x}(s)-{x}(s^{\prime })\) , \(\hat{{R}}(s,s^{\prime })={R}(s,s^{\prime })/|{R}(s,s^{\prime })|\) , and \(c = -2\log (\epsilon )\)  being a function of the slenderness parameter. This method was initially developed by [1]} and later improved by [2]}, [3]}, and [4]}. The non-local contribution (\({K}[{f}]\)  together with the \(c\) -independent part of \({\Lambda }[{f}]\) ) vanishes with comparison to the local one at a rate \(o(1/\log (\epsilon ))\) . In this contribution, we shall take advantage of this asymptotic behaviour by neglecting \({K}[{f}]\)  entirely. For slender fibres, neglecting the non-local term is a great simplification towards an analytical treatment of the resulting equations and leads to the local slender-body theory, also known as resistive-force theory (RFT) of [5]}, in which the local velocity (in a quiescent fluid) is related to the local hydrodynamic drag force on the filament, \({f}_\text{h}\) , by
\({u}(s) = -\frac{c}{8 \pi \mu }\left(\mathbb {I} + {\partial _s{x}}\,{\partial _s{x}}\right)\cdot {f}_\text{h}(s). \) 
